It turns out that \({\mathcal {E}}_s\)  is smooth and has genus 1 for complex \(s \ne 0, 1,c^2,\infty \) . The curve \({\mathcal {E}}_s\)  has a Jacobian, which is an elliptic curve (see [1]}), and may be seen as the connected component of the identity in the group of automorphisms of \({\mathcal {E}}_s\)  (they have no fixed points except for the identity). If we let \(s\)  vary arbitrarily in \({C}\) , the field of definition is immaterial, but if we consider \(s\)  as a variable, then the ground field of the whole construction may be taken as \({Q}(c,s)\) . This will be relevant in some verifications below.
